scholarly journals Trees with unique minimum glolal offensive alliance

2020 ◽  
Author(s):  
Mohamed Bouzefrane ◽  
Isma Bouchemakh ◽  
Mohamed Zamime ◽  
Noureddine Ikhlef-Eschouf
Keyword(s):  
Dominating Set ◽  
Simple Graph ◽  
Minimum Cardinality ◽  
Unique Minimum

Let G= (V,E) be a simple graph. A non-empty set D⊆V is called a global offensive alliance if D is a dominating set and for every vertex v in V-D, |N_{G}[v]∩D|≥|N_{G}[v]-D|. The global offensive alliance number is the minimum cardinality of a global offensive alliance in G. In this paper, we give a constructive characterization of trees having a unique minimum global offensive alliance.

Super domination in trees

2021 ◽  
pp. 2150137
Author(s):  
B. Senthilkumar ◽  
Y. B. Venkatakrishnan ◽  
H. Naresh Kumar
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Minimum Cardinality

Let [Formula: see text] be a simple graph. A set [Formula: see text] is called a super dominating set if for every vertex [Formula: see text], there exist [Formula: see text] such that [Formula: see text]. The minimum cardinality of a super dominating set of [Formula: see text], denoted by [Formula: see text], is called the super domination number of graph [Formula: see text]. Characterization of trees with [Formula: see text] is presented.

Outer-paired domination in graphs

2020 ◽  
Vol 12 (06) ◽  
pp. 2050072
Author(s):  
A. Mahmoodi ◽  
L. Asgharsharghi
Keyword(s):  
Perfect Matching ◽  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Minimum Cardinality ◽  
Sharp Bounds ◽  
Paired Domination Number ◽  
Vertex Set ◽  
Paired Domination ◽  
Domination In Graphs

Let [Formula: see text] be a simple graph with vertex set [Formula: see text] and edge set [Formula: see text]. An outer-paired dominating set [Formula: see text] of a graph [Formula: see text] is a dominating set such that the subgraph induced by [Formula: see text] has a perfect matching. The outer-paired domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of an outer-paired dominating set of [Formula: see text]. In this paper, we study the outer-paired domination number of graphs and present some sharp bounds concerning the invariant. Also, we characterize all the trees with [Formula: see text].

scholarly journals Graphs with Constant Sum of Domination and Inverse Domination Numbers

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
T. Asir
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Complete Characterization ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Vertex Set ◽  
Domination Numbers

A subset D of the vertex set of a graph G, is a dominating set if every vertex in V−D is adjacent to at least one vertex in D. The domination number γ(G) is the minimum cardinality of a dominating set of G. A subset of V−D, which is also a dominating set of G is called an inverse dominating set of G with respect to D. The inverse domination number γ′(G) is the minimum cardinality of the inverse dominating sets. Domke et al. (2004) characterized connected graphs G with γ(G)+γ′(G)=n, where n is the number of vertices in G. It is the purpose of this paper to give a complete characterization of graphs G with minimum degree at least two and γ(G)+γ′(G)=n−1.

scholarly journals ON SOME BOUNDS OF THE MINIMUM EDGE DOMINATING ENERGY OF A GRAPH

2021 ◽  
Vol 12 (4) ◽  
pp. 1394-1399
Author(s):  
A.Sharmila , Et. al.
Keyword(s):  
Characteristic Polynomial ◽  
Dominating Set ◽  
Simple Graph ◽  
Minimum Cardinality ◽  
Edge Dominating Set ◽  
Eigen Values ◽  
Vertex Set ◽  
G 12

Let G be a simple graph of order n with vertex set V= {v1, v2, ..., vn} and edge set  E = {e1, e2, ..., em}. A subset  of E is called an edge dominating set of G if every edge of  E -  is adjacent to some edge in  .Any edge dominating set with minimum cardinality is called a minimum edge dominating set [2]. Let  be a minimum edge dominating set of a graph G. The minimum edge dominating matrix of G is the m x m matrix defined by G)= , where  = The characteristic polynomial of is denoted by fn (G, ρ) = det (ρI -  (G) ).  The minimum edge dominating eigen values of a graph G are the eigen values of (G).  Minimum edge dominating energy of G is defined as                 (G) =   [12] In this paper we have computed the Minimum Edge Dominating Energy of a graph. Its properties and bounds are discussed. All graphs considered here are simple, finite and undirected.

scholarly journals Almost well-dominated bipartite graphs with minimum degree at least two

2020 ◽  
Author(s):  
Hadi Alizadeh ◽  
Didem Gözüpek
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Maximum Cardinality ◽  
Minimum Cardinality ◽  
Upper Domination Number ◽  
The Difference ◽  
Two Parameters

A dominating set in a graph $G=(V,E)$ is a set $S$ such that every vertex of $G$ is either in $S$ or adjacent to a vertex in $S$. While the minimum cardinality of a dominating set in $G$ is called the domination number of $G$ denoted by $\gamma(G)$, the maximum cardinality of a minimal dominating set in $G$ is called the upper domination number of $G$ denoted by $\Gamma(G)$. We call the difference between these two parameters the \textit{domination gap} of $G$ and denote it by $\mu_d(G) = \Gamma(G) - \gamma(G)$. While a graph $G$ with $\mu_d(G)=0$ is said to be a \textit{well-dominated} graph, we call a graph $G$ with $\mu_d(G)=1$ an \textit{almost well-dominated} graph. In this work, we first establish an upper bound for the cardinality of bipartite graphs with $\mu_d(G)=k$, where $k\geq1$, and minimum degree at least two. We then provide a complete structural characterization of almost well-dominated bipartite graphs with minimum degree at least two. While the results by Finbow et al.~\cite{domination} imply that a 4-cycle is the only well-dominated bipartite graph with minimum degree at least two, we prove in this paper that there exist precisely 31 almost well-dominated bipartite graphs with minimum degree at least two.

scholarly journals ON THE ROOTS OF TOTAL DOMINATION POLYNOMIAL OF GRAPHS, II

2019 ◽  
pp. 659
Author(s):  
Saeid Alikhani ◽  
Nasrin Jafari
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Total Domination ◽  
Dominating Sets ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Domination Polynomial

Let $G = (V, E)$ be a simple graph of order $n$. A  total dominating set of $G$ is a subset $D$ of $V$, such that every vertex of $V$ is adjacent to at least one vertex in  $D$. The total domination number of $G$ is  minimum cardinality of  total dominating set in $G$ and is denoted by $\gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=\sum_{i=\gamma_t(G)}^n d_t(G,i)$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. In this paper, we study roots of the total domination polynomial of some graphs.  We show that  all roots of $D_t(G, x)$ lie in the circle with center $(-1, 0)$ and radius $\sqrt[\delta]{2^n-1}$, where $\delta$ is the minimum degree of $G$. As a consequence, we prove that if $\delta\geq \frac{2n}{3}$,  then every integer root of $D_t(G, x)$ lies in the set $\{-3,-2,-1,0\}$.

scholarly journals DOMINATION AND EDGE DOMINATION IN TREES

2020 ◽  
Vol 6 (1) ◽  
pp. 147
Author(s):  
B. Senthilkumar ◽  
Yanamandram B. Venkatakrishnan ◽  
H. Naresh Kumar
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Minimum Cardinality ◽  
Edge Dominating Set

Let \(G=(V,E)\) be a simple graph. A set \(S\subseteq V\) is a dominating set if every vertex in \(V \setminus S\) is adjacent to a vertex in \(S\). The domination number of a graph \(G\), denoted by \(\gamma(G)\) is the minimum cardinality of a dominating set of \(G\). A set \(D \subseteq E\) is an edge dominating set if every edge in \(E\setminus D\) is adjacent to an edge in \(D\). The edge domination number of a graph \(G\), denoted by \(\gamma'(G)\) is the minimum cardinality of an edge dominating set of \(G\). We characterize trees with  domination number equal to twice edge domination number.

scholarly journals Minimum Total Dominating Energy of Some Standard Graphs

2020 ◽  
Vol 8 (4S5) ◽  
pp. 272-276
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Total Domination ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Ladder Graph ◽  
Eigen Values ◽  
Vertex Set

Let G be a simple graph with vertex set V(G) and edge set E(G). A set S of vertices in a graph 𝑮(𝑽,𝑬) is called a total dominating set if every vertex 𝒗 ∈ 𝑽 is adjacent to an element of S. The minimum cardinality of a total dominating set of G is called the total domination number of G which is denoted by 𝜸𝒕 (𝑮). The energy of the graph is defined as the sum of the absolute values of the eigen values of the adjacency matrix. In this paper, we computed minimum total dominating energy of a Friendship Graph, Ladder Graph and Helm graph. The Minimum total dominating energy for bistar graphand sun graph is also determined.

scholarly journals A Comparison on the Bounds of Chromatic Preserving Number and Dom-Chromatic Number of Cartesian Product and Kronecker Product of Paths

2012 ◽  
Vol 11 (4) ◽  
pp. 43-58
Author(s):  
T N Janakiraman ◽  
M Poobalaranjani
Keyword(s):  
Comparative Study ◽  
Chromatic Number ◽  
Kronecker Product ◽  
Dominating Set ◽  
Cartesian Product ◽  
Simple Graph ◽  
Minimum Cardinality ◽  
Vertex Set

Let G be a simple graph with vertex set V and edge set E. A Set S Í V is said to be a chromatic preserving set or a cp-set if χ(<S>) = χ(G) and the minimum cardinality of a cp-set in G is called the chromatic preserving number or cp-number of G and is denoted by cpn(G). A cp-set of cardinality cpn(G) is called a cpn-set. A subset S of V is said to be a dom- chromatic set (or a dc-set) if S is a dominating set and χ(<S>) = χ(G). The minimum cardinality of a dom-chromatic set in a graph G is called the dom-chromatic number (or dc- number) of G and is denoted by γch(G). The Kronecker product G1 Ù G2 of two graphs G1 = (V1, E1) and G2 = (V2, E2) is the graph G with vertex set V1 x V2 and any two distinct vertices (u1, v1) and (u2, v2) of G are adjacent if u1u2 Î E1 and v1v2 Î E2. The Cartesian product G1 x G2 is the graph with vertex set V1 x V2 where any two distinct vertices (u1, v1) and (u2, v2) are adjacent whenever (i) u1 = u2 and v1v2 Î E2 or (ii) u1u2 Î E1 and v1 = v2. These two products have no common edges. They are almost like complements but not exactly. In this paper, we discuss the behavior of the cp-number and dc-number and their bounds for product of paths in the two cases. A detailed comparative study is also done.

Edge-vertex domination in trees

2021 ◽  
Author(s):  
Kijung Kim
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Minimum Cardinality ◽  
Number Formula

Let [Formula: see text] be a finite simple graph. A vertex [Formula: see text] is edge-vertex dominated by an edge [Formula: see text] if [Formula: see text] is incident with [Formula: see text] or [Formula: see text] is incident with a vertex adjacent to [Formula: see text]. An edge-vertex dominating set of [Formula: see text] is a subset [Formula: see text] such that every vertex of [Formula: see text] is edge-vertex dominated by an edge of [Formula: see text]. The edge-vertex domination number [Formula: see text] is the minimum cardinality of an edge-vertex dominating set of [Formula: see text]. In this paper, we prove that [Formula: see text] for every tree [Formula: see text] of order [Formula: see text] with [Formula: see text] leaves, and we characterize the trees attaining each of the bounds.

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